Processing scalar implicature: a constraint-based approach.

Cognitive Science 39 (2015) 667–710 Copyright © 2014 Cognitive Science Society, Inc. All rights reserved. ISSN: 0364-0213 print / 1551-6709 online DOI: 10.1111/cogs.12171

Processing Scalar Implicature: A Constraint-Based Approach Judith Degen,a Michael K. Tanenhausb a

Department of Psychology, Stanford University Department of Brain and Cognitive Sciences, University of Rochester

b

Received 12 May 2012; received in revised form 8 July 2013; accepted 31 January 2014

Abstract Three experiments investigated the processing of the implicature associated with some using a “gumball paradigm.” On each trial, participants saw an image of a gumball machine with an upper chamber with 13 gumballs and an empty lower chamber. Gumballs then dropped to the lower chamber and participants evaluated statements, such as “You got some of the gumballs.” Experiment 1 established that some is less natural for reference to small sets (1, 2, and 3 of the 13 gumballs) and unpartitioned sets (all 13 gumballs) compared to intermediate sets (6–8). Partitive some of was less natural than simple some when used with the unpartitioned set. In Experiment 2, including exact number descriptions lowered naturalness ratings for some with small sets but not for intermediate size sets and the unpartitioned set. In Experiment 3, the naturalness ratings from Experiment 2 predicted response times. The results are interpreted as evidence for a ConstraintBased account of scalar implicature processing and against both two-stage, Literal-First models and pragmatic Default models. Keywords: Pragmatics; Scalar implicature; Quantifiers; Alternatives

1. Introduction Successful communication requires readers and listeners (hereafter, listeners) to infer a speaker’s intended meaning from an underspecified utterance. To this end listeners make use of the semantic and pragmatic information available in the linguistic input and the discourse context. However, it is an open question whether semantic information takes a privileged position in this reasoning process. In recent years, scalar implicatures as in (1) have served as an important testing ground for examining the time course of integration of semantic and pragmatic information in inferring a speaker’s intended meaning. Correspondence should be sent to Judith Degen, Department of Psychology, Stanford University, 450 Serra Mall, Stanford, CA 94305. E-mail:[email protected]

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J. Degen, M. K. Tanenhaus / Cognitive Science 39 (2015)

(1) You got some of the gumballs. ↪You got some, but not all, of the gumballs The semantic content of the utterance in (1) is that the listener got at least one gumball (lower-bound meaning of some). Under some conditions, however, the listener is also justified in taking the speaker to implicate that he, the listener, did not get all of the gumballs (upper-bound meaning of some). This is a case of scalar implicature. The term “scalar” is used because the inference that gives rise to the upper-bound interpretation is assumed to depend upon a highly accessible set of alternatives that are ordered on a scale by asymmetrical entailment (Horn, 2004), and that the speaker could have selected but did not (e.g., ). 1.1. Overview The current article focuses on how listeners compute the upper-bound interpretation, sometimes called pragmatic some. We first discuss the classic distinction between Generalized and Particularized Conversational Implicature, originally introduced by Grice (1975) to distinguish relatively context-dependent from relatively context-independent inferences. We then describe two processing hypotheses, the Default hypothesis and the two-stage, Logical/Semantic/Literal-First hypothesis, that have played a central role in guiding experimental investigations of how listeners arrive at an interpretation of pragmatic some. In doing so, we flesh out some of the assumptions that underlie these approaches. We propose a probabilistic Constraint-Based framework in which naturalness and availability of alternatives (along with other factors) play a central role in computing pragmatic some. We then focus on two superficially similar visual world experiments that test the two-stage hypothesis, finding strikingly different data patterns and arriving at opposite conclusions. We argue that the differences arise because of the naturalness and availability of other lexical alternatives to some besides the stronger scalar alternative all, in particular the effects of intermixing some and exact number with small set sizes. We test this claim in three experiments using a “gumball” paradigm. 1.2. Generalized and Particularized Conversational Implicatures According to Grice (1975), scalar implicatures are an instance of Generalized Conversational Implicature (GCI). These are implicatures that are assumed to arise in the same systematic way regardless of context. In the case of scalar implicature, the speaker is taken to convey the negation of a stronger alternative statement that she could have made (and that would have also been relevant) but chose not to. Instead of (1), the speaker could have said, You got all of the gumballs, which would have also been relevant and more informative. Together with the additional assumption that the speaker is an authority on whether or not the stronger statement is true (the Competence Assumption; Sauerland, 2004; Van Rooij & Schulz, 2004), the listener may infer


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that the speaker is conveying that the listener in fact got some, but not all, of the gumballs. Generalized Conversational Implicatures are distinguished from Particularized Conversational Implicatures (PCI), which are strongly dependent on specific features of the context, such as the nature of the Question Under Discussion (QUD; Roberts, 1996). For example, if the sentence in (1) was uttered in a context in which the listener—say, a little boy—was complaining that he did not get any candy, the speaker—say, his mother—may be taken to implicate that he should stop complaining. However, this implicature does not arise if the context is modified slightly; for example, if uttered in a context where the child cannot see which objects came out of a prize machine and asks his parents whether he got all of the gumballs, the scalar implicature that he did not get all of the gumballs will presumably arise, whereas the implicature that he should stop complaining does not. Although it is sometimes claimed that scalar implicatures arise in nearly all contexts (Levinson, 2000), scalar implicatures can, in fact, be canceled not only explicitly as in (2a) but also implicitly as in (2b). (2a) You got some of the gumballs. In fact, you got all of them. (2b) A: Did I get anything from the prize machine? B: You got some of the gumballs. In (2a), the implicature is explicitly canceled because it is immediately superseded by assertion of the stronger alternative. In (2b), it is canceled implicitly because the preceding context makes the stronger alternative with all irrelevant. What matters to A in (2b) is not whether he got all versus some but all of the gumballs, but whether he got any items at all (e.g., gumballs, rings, smarties) from the machine. This feature of implicit cancelability has played a crucial role in experimental studies of implicatures (e.g., Bott & Noveck, 2004; Breheny, Katsos, & Williams, 2006) because hypotheses about the time course of scalar implicature processing make different claims about the status of implicit implicature cancelation. 1.3. Processing frameworks Formal accounts of conversational implicatures typically do not specify how a listener might compute an implicature as an utterance unfolds in a specific conversational context (e.g., Gazdar, 1979; Horn, 1984). Recently, however, there has been increased interest in how listeners compute scalar implicatures in real-time language comprehension. Most of this work has focused on distinguishing between two alternative hypotheses. The first hypothesis is that implicatures are computed immediately and effortlessly due to their status as default inferences (Levinson, 2000; see also Chierchia, 2004 for a related account under which scalar implicatures are taken to arise by default in

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upward-entailing, but not in downward-entailing contexts). Under this Default account, scalar implicatures do not incur the usual processing costs associated with generating an inference. Only cancelation of an implicature takes time and processing resources. This view is motivated by considerations about communicative efficiency. Rapid, effortless inference processes for deriving GCIs are proposed as a solution to the problem of the articulatory bottleneck: While humans can only produce a highly limited number of phonemes per second, communication nevertheless proceeds remarkably quickly. The inferences that allow listeners to derive context-dependent interpretations, such as those presumably involved in computing particularized implicatures, are assumed to be slow and resource-intensive, in the sense of classic two-process models of attention that distinguish between automatic processes, which are fast, require few resources, and arise independent of context, and controlled processes, which are slow, strategic, and resource demanding (Neely, 1977; Posner & Snyder, 1975; Shiffrin & Schneider, 1977; also see Kahneman, 2011 for a related framework). In contrast, a scale containing a small set of lexical alternatives—like —could be pre-compiled and thus automatically accessed, regardless of context. Because the default interpretation arises automatically, there will be a costly interpretive garden-path when the default meaning has to be overridden. Therefore, the default model predicts that in all contexts, a default, upper-bound interpretation will precede a possible lower-bound interpretation. The second hypothesis, also termed the Literal-First hypothesis (Huang & Snedeker, 2009), assumes that the lower-bound semantic interpretation is computed rapidly, and perhaps automatically, as a by-product of basic sentence processing. All inferences, including GCIs, require extra time and resources. Therefore, the Literal-First hypothesis predicts that in all contexts a lower-bound interpretation will be computed before an upper-bound interpretation is considered. For proponents of the Literal-First hypothesis this follows from the traditional observation in the linguistic literature (e.g., Horn, 2004) that the semantic interpretation of simple declaratives containing some is in an important sense more basic than the pragmatic interpretation: The upper-bound interpretation some but not all always entails the lower-bound interpretation at least one.1 The pragmatic interpretation cannot exist without the semantic one, which translates into a two-stage processing sequence: Upon encountering a scalar item like some, the semantic interpretation is necessarily constructed before the pragmatic one. To the extent that there is a processing distinction between Generalized and Particularized Implicatures, it is that the relevant dimensions of the context and the interpretations that drive the inference are more circumscribed and thus perhaps more accessible for Generalized Implicatures. The Default and two-stage models make straightforward predictions about the time course of the implicature associated with the upper-bound interpretation of utterances containing some. The Default hypothesis predicts that logical/semantic some should be more resource-intensive and slower than upper-bound, pragmatic some, whereas the two-stage model makes the opposite prediction. There are, however, alternative Context-Driven frameworks in which the speed with which a scalar implicature is computed is largely determined by context (e.g., Breheny


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et al., 2006). For example, Relevance Theory (Sperber & Wilson, 1995), like the Literal-First hypothesis, assumes that the semantic interpretation is basic. In Relevance Theory, the upper-bound meaning is only computed if required to reach a certain threshold of relevance in context. In contrast to the Literal-First hypothesis, however, Relevance Theory does not necessarily assume a processing cost for the pragmatic inference. If the context provides sufficient support for the upper-bound interpretation, it may well be computed without incurring additional processing cost. However, a processing cost will be incurred if the upper-bound interpretation is relevant but the context provides less support. The Constraint-Based framework, which guides our research, is also Context-Driven. It falls most generally into the class of generative (i.e., data explanation) approaches to perception and action (Clark, 2013) which view perception as a process of probabilistic, knowledge-driven inference. Our approach is grounded in three related observations. First, as an utterance unfolds, listeners rapidly integrate multiple sources of information. That is, utterance comprehension is probabilistic and constraint-based (MacDonald, Pearlmutter, & Seidenberg, 1994; Seidenberg & Macdonald, 1999; Tanenhaus & Trueswell, 1995). Second, listeners generate expectations of multiple types about the future, including the acoustic/phonetic properties of utterances, syntactic structures, referential domains, and possible speaker meaning (Chambers, Tanenhaus, & Magnuson, 2004; Chang, Dell, & Bock, 2006; Kutas & Hillyard, 1980; Levy, 2008; Tanenhaus, SpiveyKnowlton, Eberhard, & Sedivy, 1995; Trueswell, Tanenhaus, & Kello, 1993). Third, interlocutors can rapidly adapt their expectations to different speakers, situations, etc. (Bradlow & Bent, 2008; Clayards, Tanenhaus, Aslin, & Jacobs, 2008; Fine, Jaeger, Farmer, & Qian, 2013; Grodner & Sedivy, 2011; Kurumada, Brown, & Tanenhaus, 2012). Given these assumptions, standard hierarchical relations among different types of representations need not map onto temporal relationships in real-time processing. We illustrate these points with three examples grounded in recent results in the language processing literature. The first example involves the mapping between speech perception, spoken word recognition, parsing, and discourse. Consider a spoken utterance that begins with “The lamb. . .” As the fricative at the onset of the unfolds, the acoustic signal provides probabilistic evidence for possible words, modulated by the prior expectations of these possible words, for example, their frequency, likely conditioned on the position in the utterance. Thus, after hearing only the first 50–100 ms of the signal, the listener has partial evidence that “the” is likely to be the word currently being uttered. Put differently, the hypothesis that the speaker intended to say the begins to provide an explanation for the observed signal. In addition, there is evidence for a likely syntactic structure (noun phrase) and evidence that the speaker intends (for some purpose) to refer to a uniquely identifiable entity (due to presuppositions associated with the definite determiner the, Russell, 1905). Thus, referential constraints can be available to the listener before any information in the speech signal associated with the production of the noun is available. The second example focuses on the interpretation of referential expressions in context. Consider the instruction in (4):

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(4) Put the pencil below the big apple. Assume that the addressee is listening to the utterance in the context of a scene with a large apple, a smaller apple, a large towel, and a small pencil. According to standard accounts of scalar adjectives, a scalar dimension can only be interpreted with respect to a reference class given by the noun it modifies (e.g., what counts as big for a building differs from what counts as big for a pencil). Translating this directly into processing terms, when the listener encounters the scalar adjective big, interpretation should be delayed because the reference class has not yet been established. However, in practice the listener’s attention will be drawn to the larger of the two apples before hearing the word apple because use of a scalar adjective signals a contrast among two or more entities of the same semantic type (Sedivy, Tanenhaus, Chambers, & Carlson, 1999). Thus, apple will be immediately interpreted as the larger of the two apples. More generally, addressees circumscribe referential domains on the fly, taking into account possible actions as well as the affordances of objects (Chambers et al., 2002; Chambers et al., 2004; Tanenhaus et al., 1995). Finally, listeners rapidly adjust expectations about the types of utterances that speakers will produce in a particular situation. After brief exposure to a speaker who uses scalar adjectives non-canonically, for example, a speaker who frequently overmodifies, an addressee will no longer assume that a pre-nominal adjective will be used contrastively (Grodner & Sedivy, 2011). Likewise, the utterance, It looks like a zebra is most canonically interpreted to mean The speaker thinks it’s a zebra (with noun focus), or You might think it’s a zebra, but it isn’t (with contrastive focus on the verb), depending on the context, the prosody, and the knowledge of the speaker and the addressee (Kurumada et al., 2012). Crucially, when a stronger alternative is sometimes used by the speaker, for example, It is a zebra, then the interpretation of It looks like a zebra with noun focus is more likely to receive the but it’s not interpretation (Kurumada et al., 2012). Again an utterance is interpreted with respect to the context and the set of likely alternatives to the observed utterance. When we embed the issue of when, and how quickly, upper- and lower-bound some are interpreted within a Constraint-Based approach with expectations and adaptation, then questions about time course no longer focus on distinguishing between the claim that scalar implicatures are computed by default and the claim that they are only computed after an initial stage of semantic processing. When there is more probabilistic support from multiple cues, listeners will compute scalar inferences more quickly and more robustly. Conversely, when there is less support, listeners will take longer to arrive at the inference, and the inference will be weaker (i.e., more easily cancelable). Under the Constraint-Based account, then, the research program becomes one of identifying the cues that listeners use in service of the broader goal of understanding the representations and processes that underlie generation of implied meanings.


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1.4. Cues under investigation We focus on two types of cues, both of which are motivated by the notion that speakers could have produced alternative utterances. The first cue is the partitive of, which marks the difference between (5) and (6). (5) Alex ate some cookies. (6) Alex ate some of the cookies. While it may seem intuitively clear to the reader that (6) leads to a stronger implicature than (5), this intuition has not previously been tested. Moreover, researchers have used either one or the other form in their experimental stimuli without considering the effect this might have on processing. For example, some researchers who find delayed implicatures and use some form of the Literal-First hypothesis to explain these findings have consistently used the non-partitive form (Bott, Bailey, & Grodner, 2012; Bott & Noveck, 2004; Noveck & Posada, 2003). The absence of the partitive may provide weaker support for the implicature than use of the partitive form would have provided. Under a Constraint-Based account, this should result in increased processing effort. The second cue is the availability of lexical alternatives to some that listeners assume are available to the speaker. In general, the alternatives to some will be determined by the context in which an utterance occurs. We assume that lexical items that are part of a scale will generally be available as alternatives.2 However, in a given context, other lexical items may be introduced that become salient alternatives. In the current studies, we focus on the effects of number terms as alternatives, and the effects of intermixing some with exact number terms, for example, You got two of the gumballs versus You got some of the gumballs. The motivation for using exact number terms as a case study comes from two studies (Grodner, Klein, Carbary, & Tanenhaus, 2010; Huang & Snedeker, 2009), which used similar methods but found different results. Both Huang and Snedeker (2009, 2011) and Grodner et al. (2010) used the visual world eye-tracking paradigm (Cooper, 1974; Tanenhaus et al., 1995). In Huang and Snedeker’s (2009, 2011) experiments participants viewed a display with four quadrants, with the two left and the two right quadrants containing pictures of children of the same gender, with each child paired with objects. For example, on a sample trial, the two left quadrants might each contain a boy: one with two socks and one with nothing. The two right quadrants might each contain a girl: one with two socks (pragmatic target) and one with three soccer balls (literal target). A preamble established a context for the characters in the display. In the example, the preamble might state that a coach gave two socks to one of the boys and two socks to one of the girls, three soccer balls to the other girl, who needed the most practice, and nothing to the other boy. Participants were asked to follow instructions such as Point to the girl who has some of the socks. Huang and Snedeker (2009) reasoned that if the literal interpretation is

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computed prior to the inference, then, upon hearing some, participants should initially fixate both the semantic and pragmatic targets equally because both are consistent with the literal interpretation. If, however, the pragmatic inference is immediate, then the literal target should be rejected as soon as some is recognized, resulting in rapid fixation of the pragmatic target. The results strongly indicated that the literal interpretation was computed first. For commands with all (e.g., Point to the girl who has all of the soccer balls) and commands using number (e.g., Point to the girl who has two/three of the soccer balls), participants converged on the correct referent 200–400 ms after the quantifier. In contrast, for commands with some, target identification did not occur until 1,000– 1,200 ms after the quantifier onset. Moreover, participants did not favor the pragmatic target prior to the noun’s phonetic point of disambiguation (e.g., -ks of socks). Huang and Snedeker concluded that “even the most robust pragmatic inferences take additional time to compute” (Huang & Snedeker, 2009, p. 408). The Huang and Snedeker results complement previous response time (Bott & Noveck, 2004; Noveck & Posada, 2003) and reading-time experiments (Breheny et al., 2006). In these studies, response times associated with the pragmatic inference are longer than both response times to a scalar item’s literal meaning and to other literal controls (typically statements including all). Moreover, participants who interpret some as some and possibly all, so-called logical responders (Noveck & Posada, 2003), have faster response times than pragmatic responders who interpret some as some but not all. In contrast, Grodner et al. (2010) found evidence for rapid interpretation of pragmatic some, using displays and a logic similar to that used by Huang and Snedeker (2009). Each trial began with three boys and three girls on opposite sides of the display and three groups of objects in the center. A pre-recorded statement described the total number and type of objects in the display. Objects were then distributed among the participants. The participant then followed a pre-recorded instruction, of the form Click on the girl who has summa/nunna/alla/the balloons. Convergence on the target for utterances with some was just as fast as for utterances with all. Moreover, for trials on which participants were looking at the character with all of the objects at the onset of the quantifier summa) participants began to shift fixations away from that character and to the character(s) with only some of the objects (e.g., the girl with the balls or the girl with the balloons) within 200–300 ms after the onset of the quantifier. Thus, participants were immediately rejecting the literal interpretation as soon as they heard summa. If we compare the results for all and some in the Huang and Snedeker and Grodner et al. experiments, the time course of all is similar but the upperbound interpretation of partitive some is computed 600–800 ms later in Huang and Snedeker. Why might two studies as superficially similar as Huang and Snedeker (2009) and Grodner et al. (2010) find such dramatically different results? In Degen and Tanenhaus (under review), we discuss some of the primary differences between the two studies, concluding that only the presence or absence of number expressions might account for the conflicting results. Huang and Snedeker included stimuli with numbers, whereas Grodner


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et al. did not. In Huang and Snedeker (2009) a display in which one of the girls had two socks was equally often paired with the instruction, Point to the girl who has some of the socks and Point to the girl who has two of the socks. In fact, Huang, Hahn, and Snedeker (2010) have shown that eliminating number instructions reduces the delay between some and all in their paradigm. Why might instructions with exact number delay upper-bound interpretations of partitive some? Computation of speaker meaning takes into account what the speaker could have, but did not say, with respect to the context of the utterance. Recall that the upperbound interpretation of some is licensed when the speaker could have, but did not say all, because all would have been more informative. More generally, it would be slightly odd for a speaker to use some when all would be the more natural or typical description. We propose that in situations like the Huang and Snedeker and Grodner et al. experiments, exact number is arguably more natural than some. In fact, intuition suggests that mixing some and exact number makes some less natural. Consider a situation where there are two boys, two girls, four socks, three soccer balls, and four balloons. One girl is given two of the four socks, one boy the four balloons, and the other boy two of the three soccer balls. Assuming the speaker knows exactly who got what, the descriptions in (7) and (8) seem natural, compared to the description in (9): (7) One of the girls got some of the socks and one of the boys got all of the balloons. (8) One of the girls got two of the socks and one of the boys got all of the balloons. (9) One of the girls got two of the socks and one of the boys got some of the soccer balls. Grodner et al. (2010) provided some empirical support for these intuitions. They collected naturalness ratings for their displays and instructions, both with and without exact number included in the instruction set. Including exact number lowered the naturalness ratings for partitive some but not for all. However, even without exact number instructions, some was rated as less natural than all. One reason might be that in most situations, pragmatic some is relatively infelicitous when used to describe small sets. Again, intuition suggests that using some is especially odd for sets of one and two. Consider a situation where there are three soccer balls and John is given one ball and Bill two. John got one of the soccer balls seems a more natural description than John got some of the soccer balls and Bill got two of the soccer balls seems more natural than Bill got some of the soccer balls. These observations suggest an alternative hypothesis for why responses to pragmatic some are delayed when intermixed with exact number for small set sizes. Most generally we suggest that some will compete with other rapidly available contextual alternatives. In particular, we hypothesize that the mapping between an utterance with some and an interpretation of that utterance is delayed when there are rapidly available, more natural alternatives to describe the state of the world. This seems likely to be the case for exact

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number descriptions with small sets because the more natural number alternative is also a number in the subitizing range where number terms become rapidly available and determining the cardinality of a set does not require counting (Atkinson, Campbell, & Francis, 1976; Kaufman, Lord, Reese, & Volkman, 1949; Mandler, Shebo, & Vol, 1982). In situations where exact number is available as a description, the number term is likely to become automatically available, thus creating a more natural, more available interpretation of the scene. In Gricean terms, we are proposing that delays in response times (to press a button or to fixate a target) which have previously been argued to be due to the costly computation of the Quantity implicature from some to not all might in fact be due to interference from lexical alternatives to some. In the Huang and Snedeker studies, in particular, there may be interference from number term alternatives that, while scalar in nature, function on a different scale than some. In Gricean terms, the motivation for this interference comes from the maxim of Manner: If there is a less ambiguous quantifier (e.g., two) that could have been chosen to refer to a particular set size, the speaker should have used it. If she did not, it must be because she meant something else. Finally arriving at the implicature from some to not all, when there are more natural lexical alternatives to some that the speaker could have used but did not, thus may involve both reasoning involving the Quantity maxim (standard scalar implicature) and the Manner maxim (inference that the speaker must have not meant the partitioned set which could have more easily and naturally been referred to by two). If this is indeed the case, this would have serious implications for the interpretation of response time results on scalar implicature processing across the board; previously, delays in computing pragmatic some have been associated with the costly computation of the scalar implicature itself, taking into account only the competition of some with its scalemate all. What we are proposing here is that this view is too narrow—rather than competing only with its lexicalized scalemate, some contextually competes with many other alternatives, like number terms, which are not lexicalized alternatives. Thus, observed delays in the processing of scalar implicatures might be at least partly due to costly reasoning about unnatural, misleading quantifier choices. 1.5. The gumball paradigm The current experiments examine the role of contextual alternatives within the Constraint-Based framework using the case of exact number. Our specific hypothesis is that number selectively interferes with some—both in processing the upper-bound and the lower-bound interpretation—where naturalness of some is low and number terms are rapidly available. We evaluate this hypothesis in a series of experiments using a “gumball paradigm.” We begin with an example to illustrate how likely interpretations might change over time. Suppose that there is a gumball machine with 13 gumballs in the upper chamber. Alex knows that this gumball machine has an equal probability of dispensing 0–13 gumballs. His friend, Thomas, inserts a quarter and some number of gumballs drops to the lower chamber but Alex cannot see how many have dropped. Thomas, however, can, and he says You got some of the gumballs.


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Before the start of the utterance Alex will have certain prior expectations about how many gumballs he got—in fact, in this case there is an equal probability of 1/14 that Alex got any number of gumballs. This is shown in the first panel of Fig. 1. Once Alex hears You got some, he has more information about how many gumballs he got. First, the meaning of some constrains the set to be larger than 0. However, Alex also has knowledge about how natural it would be for Thomas to utter some instead of, for example, an exact number term, to inform him of how many gumballs he got. Fig. 1 illustrates how this knowledge might shift his subjective belief probabilities for having received specific numbers of gumballs. Alex is now more certain that he has received an intermediate set size rather than a small set (where Thomas could have easily said one or two instead of some) or a large set (where Thomas could have said most, or even all). Finally, once Alex hears the partitive of, his expectations about how many gumballs he got might shift even more (e.g., because Alex knows that the partitive is a good cue to the speaker meaning to convey upper-bound some), as shown in the third panel.3 Thus, by the end of the utterance Alex will be fairly certain that he did not get all of the gumballs, but he will also expect to have received an intermediate set size, as there would have been more natural alternative utterances available to pick out either end of the range of gumballs. Note that without additional assumptions, neither the Default nor the Literal-First model makes gradient predictions about expected set size. Under the Default model, the distribution on states would be uniform until the word some is encountered, at which point both the zero-gumball and all-gumball state would be excluded as potential candidates, leaving a uniform distribution over states 1–12. In contrast, the Literal-First model predicts that upon encountering some, only the zero-gumball state should be excluded. Both models predict that over time, the integration of further contextual information may require re-including the all-state in the set of possible states (Default), or excluding the all-state from said set (Literal-First). However, neither of these models directly predicts

Fig. 1. Possible constraint-based update of most likely intended interpretation at different incremental time points. Interpretations are represented as set sizes. Bars represent the amount of probabilistic support provided for each interpretation, given the constraints at each point.

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variability in naturalness of partitive versus non-partitive some for any set size, nor variability in the naturalness of some used with different set sizes. Moreover, adding probabilistic assumptions that would map onto the time course of initial processing of some would be inconsistent with the fundamental assumptions of each of these models. We developed a gumball paradigm based on this scenario in order to investigate whether listeners are sensitive to the partitive and to the naturalness and availability of number descriptions as lexical alternatives to some in scalar implicature processing using a range of different set sizes. On each trial the participant sees a gumball machine with an upper chamber and a lower chamber, as illustrated in Fig. 2. All of the gumballs begin in the upper chamber. After a brief delay, some number of gumballs drops to the lower chamber. The participant then responds to a statement describing the scene, either by rating the statement’s naturalness or judging whether they agree or disagree with the statement. We can thus obtain information about participants’ judgments while at the same time recording response times as a measure of their interpretation of different quantifiers with respect to different visual scenes. The rationale for the rating studies is that we need to establish the relative naturalness of alternative descriptions for particular set sizes. These data are crucial for generating time course predictions that distinguish the Default, Literal-First, and Constraint-Based approaches. Crucially, naturalness data are essential for evaluating the time course claims of the Constraint-Based approach. In two rating studies in which the upper chamber begins with 13 gumballs, we establish the naturalness of some for different set sizes of interest, in particular small sets (1–3), intermediate sets (6–8), and for the unpartitioned set (all 13 gumballs). In addition, we investigate the relative naturalness of simple some versus partitive some of for the unpartitioned set. We further investigate the effect of including exact number descriptions on naturalness ratings for some and some of used

Fig. 2. Sample displays in the gumball paradigm. Left: initial display. Right: sample second display with dropped gumballs.


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with different set sizes in Experiment 2. These results allow us to make specific predictions about response times that are tested in Experiment 3.

2. Experiment 1 Experiment 1 was conducted to determine the naturalness of descriptions with some, some of (henceforth summa), all of (henceforth all), and none of (henceforth none) for set sizes ranging from 0 to 13. 2.1. Methods 2.1.1. Participants Using Amazon’s Mechanical Turk, 120 workers were paid $0.30 to participate. All were native speakers of English (as per requirement) who were na€ıve as to the purpose of the experiment. 2.1.2. Procedure and materials On each trial, participants saw a display of a gumball machine with an upper chamber filled with 13 gumballs and an empty lower chamber (Fig. 2). After 1.5 s, a new display was presented in which a certain number of gumballs had dropped to the lower chamber. Participants heard a pre-recorded statement of the form You got X gumballs, where X was a quantifier. They were then asked to rate how naturally the scene was described by the statement on a seven-point Likert scale, where seven was very natural and one was very unnatural. If they thought the statement was false, they were asked to click a FALSE button located beneath the scale. We varied both the size of the set in the lower chamber (0–13 gumballs) and the quantifier in the statement (some, summa, all, none). Some trials contained literally false statements. For example, participants might get none of the gumballs and hear You got all of the gumballs. These trials were interspersed in order to have a baseline against which to compare naturalness judgments for some (of the) used with the unpartitioned set. If interpreted semantically (as You got some and possible all of the gumballs), the some statement is true (however unnatural) for the unpartitioned set. However, if it is interpreted pragmatically as meaning You got some but not all of the gumballs, it is false and should receive a FALSE rating. Participants were assigned to one of 24 lists. Each list contained six some trials, six summa trials, two all trials, and two none trials. To avoid an explosion of conditions, each list sampled only a subset of the full range of gumball set sizes in the lower chamber. The quantifiers some and summa occurred once each with 0 and 13 gumballs. In addition, none occurred once (correctly) with 0 gumballs and all once (correctly) with 13 gumballs. Each of all and none also occurred once with an incorrect number of gumballs. The remaining some and summa trials sampled two data points each per quantifier from the subitizing range (1–4 gumballs), one from the mid range (5–8 gumballs), and one from the high range (9–12 gumballs). For an overview, see Table 1. See Appendix A,

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Table A1, for the set sizes that were sampled on each list. From each of 12 base lists, one version used a forward order and the other a reverse order. Five participants were assigned to each list. 2.2. Results and discussion Mean ratings for each quantifier for the different set sizes are presented in Fig. 3. Clicks of the FALSE button were coded as 0. This was motivated by many participants’ use of the lowest point on the scale to mean FALSE (see the histogram of responses to some/summa in Fig. 4). We thus treated all responses on a single, continuous dimension. Mean ratings were 5.83 for none (0 gumballs) and 6.28 for all (13 gumballs) and close to Table 1 Distribution of the 16 experimental trials over quantifiers and set sizes Set Size Quantifier

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J. Degen, M. K. Tanenhaus / Cognitive Science 39 (2015) numbers absent (Exp. 1)

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0 otherwise. Mean ratings for some/summa were lowest for 0 gumballs (1.4/1.1), increased to 2.7 and 5.2 for 1 and 2 gumballs, respectively, peaked in the mid range (5.81/5.73), decreased again in the high range (4.69/4.74), and decreased further at the unpartitioned set (3.47/2.7).

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Our analyses were designed to address the following three questions: (a) for which set size ranges some and summa are deemed most natural; (b) whether some and summa differ in naturalness, and (c) whether the naturalness of some and summa differs for different set sizes (or ranges of set sizes). The data were analyzed using a series of mixed effects linear regression models with byparticipant random intercepts to predict ratings. p-values were obtained using MCMC sampling (Baayen, Davidson, & Bates, 2008). For some/summa used with each unique set size in the lower chamber (0–13), we fit one model each to each subset of the data corresponding to that set size in addition to trials where all and none were used with their correct set size (13 and 0, respectively). Each model included a centered fixed effect of quantifier (some/ summa vs. all/none). Some and summa were most natural in the mid range (numerically peaked when used with six gumballs), where ratings did not differ from ratings for none and all used with their correct set size (b = 0.2, SE = 0.24, t = 0.81, p < .42). Mean ratings for some and summa did not differ for any set size except at the unpartitioned set, where some was more natural than summa (b = 0.77, SE = 0.19, t = 4.07, p < .01). This naturalness difference between some and summa used with the unpartitioned set suggests that summa is more likely to give rise to a scalar implicature than some and is thus dispreferred with the unpartitioned set. Because ratings for some and summa did not differ anywhere except for the unpartitioned set, we henceforth report collapsed results for some and summa. To test the hypothesis that naturalness for some varies with set size, we fit a mixed effects linear model predicting mean naturalness rating from range (subitizing [one to four gumballs] or mid range [five to eight gumballs]) to the subset of the some cases in each range. As predicted, naturalness was lower in the subitizing range than in the mid range (b = 0.79, SE = 0.13, t = 6.01, p < .001). Similarly, naturalness ratings for some were lower for the unpartitioned set than in the mid range (b = 2.68, SE = 0.15, t = 17.82, p < .001). However, Fig. 3 suggests that the naturalness ratings differed for set sizes within the subitizing range (1–4). Performing the analysis on subsets of the data comparing each set size with naturalness of some/summa used with the preferred set size (six gumballs) yields the following results. Collapsing over some and summa, we coded small set versus six gumballs as 0 and 1, respectively, and subsequently centered the predictor. The strongest effect is observed for one gumball (b = 2.95, SE = 0.17, t = 14.28, p < .0001). The effect is somewhat weaker for two (b = 1.0, SE = 0.22, t = 4.47, p < .0001), even weaker for three (b = 0.36, SE = 0.2, t = 1.81, p = .05), and only marginally significant for four (b = 0.29, SE = 0.19, t = 1.52, p < .1).4 The coefficients for the set size predictor for each subset model are plotted in Fig. 5. Given these results we will refer to “small set size” effects rather than “subitizing” effects. In sum, some and summa were both judged to be quite unnatural for set sizes of 1 and 2, and more natural but not quite as natural as for the preferred set size (six gumballs) for 3. Naturalness also decreased after the mid range (five to eight gumballs) and was low at the unpartitioned set. In addition, the partitive, some of, was less natural to refer to the unpartitioned set than simple some. Finally, we note that naturalness ratings for some/summa gradually decreased for set sizes above 6. This is probably due to there being other more natural,


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salient alternatives for that range: many and most. It is striking that these alternatives seem to affect the naturalness of some just as much as number terms for small set sizes. The naturalness results from this study point to an interesting fact about the meaning of some. The linguistic literature standardly treats the semantics of some as proposed in Generalized Quantifier Theory (Barwise & Cooper, 1981) as the existential operator (corresponding to at least one). Under this view, as long as at least one gumball dropped to the lower chamber, participants should have rated the some statements as true (i.e., not clicked the FALSE button). However, this was not the case: some received 12% FALSE ratings for one gumball and 9% FALSE ratings for the unpartitioned set; summa statements were rated FALSE in 7% of cases for one gumball and 18% of cases for the unpartitioned set. For comparison, rates of FALSE responses to some/summa for all other correct set sizes were 0%. In addition, treating some as simply the existential operator does not allow a role for the naturalness of quantifiers. What matters is that a statement with some is true or false. Differences in naturalness are not predicted. Whether this means that language users’ underlying representation of some is more complex than the existential operator (and similarly for other quantifiers) is an open question. One could argue for an analogy to the distinction between the underlying category and what affects categorization (for discussion of this perspective, see Armstrong, Gleitman, & Gleitman, 1983). However, our preferred view is that for the purposes of formalizing truth conditions, the existential operator is useful as an abstraction over the possible contexts in which a simple statement with some could be true. The underlying cognitive representations, on the other hand, are likely to involve mappings onto expectations of usage in specific contexts.

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One way of conceptualizing these naturalness results is that we have obtained probability distributions over set sizes for different quantifiers, where the relevant probabilities are participants’ subjective probabilities of expecting a speaker to use a particular quantifier with a particular set size. Thus, a vague quantifier like some, where naturalness is high for intermediate sets and gradually drops off at both ends of the spectrum, has a very wide distribution, with probability mass distributed over many set sizes. In contrast, for a number term like two, one would expect naturalness to be very high for a set size of two and close to 0 for all other cardinalities, and thus the distribution would be very narrow and peaked around 2. According to the Constraint-Based account that allows for parallel processing of multiple sources of information, distributions of quantifiers over set sizes (or in other words listeners’ expectations about speakers’ quantifier use) are a function of at least two factors: (a) set size and (b) awareness of contextual availability of alternative quantifiers. If no lexical alternative is available, listeners will have some expectations about the use of some with different set sizes. We propose that the distribution of ratings obtained in Experiment 1 reflects just these expectations. For some, naturalness is highest for intermediate set sizes and drops off at both ends of the tested range. That is, listeners’ expectation for some to be used is highest in the mid range. The Constraint-Based account predicts that listeners’ expectations about quantifier use are sensitive to alternatives. Including number terms among the experimental items, thus making participants aware that number terms are contextually available alternatives to some, should change this distribution. In particular, the prediction is that due to subitizing processes, which allow number terms to become rapidly available as labels for small sets, the naturalness of some should decrease for small sets when number terms are included. In other words, participants’ expectations that a small set will be referred to by some should decrease. This prediction is tested in Experiment 2.

3. Experiment 2 Experiment 2 tested the hypothesis that when number terms are included as alternatives within the context of an experiment, the naturalness of some will be reduced when it is used with small set sizes. Using the same paradigm as in Experiment 1, we included number terms among the stimuli to test the hypothesis that the naturalness of some/ summa would be reduced when used with small set sizes, where number terms are hypothesized to be most natural. 3.1. Methods 3.1.1. Participants Using Amazon’s Mechanical Turk, 240 workers were paid $0.75 to participate. All were native speakers of English (as per requirement) who were na€ıve as to the purpose of the experiment.


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3.1.2. Procedure and materials The procedure was the same as that described for Experiment 1 with one difference; the number terms one of the through twelve of the were included among the stimuli. Each participant rated naturalness of statements with quantifiers as descriptions of gumball machine scenes on 32 trials. Participants were assigned to one of 48 lists. As in Experiment 1, each list contained six some trials, six summa trials, two all trials, and two none trials (see Table 1 for an overview of these 16 trials). In addition, four number terms were included on each list. Each number term occurred once with its correct set size, once with a wrong set size that differed from the correct set size by one gumball, and once with a wrong set size that differed from the correct set size by at least three gumballs. The lists were created from the same base lists used in Experiment 1. See Appendix A for the set sizes that were sampled on each list. Four versions of each of the twelve base lists were created. On half of the lists, some/summa occurred before the correct number term for each set size; on the other half, it occurred after the correct number term. Half of the lists sampled wrong set sizes that were one bigger than the correct set size for the number terms employed, and half sampled set sizes that were one smaller. 3.2. Results As in Experiment 1, clicks of the FALSE button were coded as 0 and ratings treated as continuous values (see Fig. 4 for a histogram of the distribution of ratings for each number of gumballs). Mean ratings were 5.71 for none with 0 gumballs and 6.31 for all with 13 gumballs, and close to 0 otherwise. Mean ratings for some/summa were lowest for 0 gumballs (1.08/0.96), increased from 2.02/1.99 to 4.91/4.54 in the small set range, peaked in the mid range at six gumballs (5.82/5.97), decreased again in the high range (4.33/4.47), and decreased further at the unpartitioned set (3.42/2.65), replicating the general shape of the curve obtained in Experiment 1. Again, some and summa were most natural when used with six gumballs. As in Experiment 1, mean ratings for some and summa did not differ for any set size except at the unpartitioned set, where some was more natural than summa (b = 0.77, SE = 0.13, t = 4.07, p < .001). To test the hypothesis that adding number terms decreases the naturalness for some/ summa in the small set range but nowhere else, we fit a series of mixed effects linear models with random by-participant intercepts, predicting mean naturalness rating from number term presence for each range (no gumballs, small, mid, high, unpartitioned set). The models were fit to the combined data sets from Experiment 1 (numbers absent) and Experiment 2 (numbers present). As predicted, naturalness was lower for both some and summa when numbers were present in the small set range (b = 0.49, SE = 0.16, t = 3.13, p < .002), but not for 0 gumballs (b = 0.19, SE = 0.17, t = 1.11, p < .28), in the mid range (b = 0.14, SE = 0.14, 0.99, p = .13), in the high range (b = 0.38, SE = 0.23, t = 1.37, p < .12), or with the unpartitioned set (b = 0.11, SE = 0.23, t = 0.49, p < .71). Fig. 6 presents the mean naturalness ratings when numbers were present (Experiment 2) and absent (Experiment 1).

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Performing each analysis individually for each set size in the subitizing range shows that the strength of the number presence effect in the small set range differed for different set sizes. It was strongest for one gumball (b = 0.83, SE = 0.27, t = 3.12, p < .0001), less strong for two (b = 0.54, SE = 0.22, t = 2.4, p < .01), trending for three gumballs (b = 0.27, SE = 0.2, t = 1.16, p < .12), and nonsignificant for four gumballs (b = 0.1, SE = 0.18, t = 0.57, p < .61). We provide a coefficient plot for the effect of number presence for different set sizes in Fig. 7. Therefore, naturalness effects are not due to subitizing per se, as we had initially hypothesized. Rather, subitizing might interact with naturalness to determine the degree to which number alternatives compete with some. Number terms did not reduce naturalness for none (b = 0.1, SE = 0.22, t = 0.47, p < .6) and all (b = 0.02, SE = 0.17, t = 0.14, p < .84) when used with their correct set sizes. Finally, although ratings for number term ratings are extremely high throughout when used with their correct set size and close to floor otherwise, number terms are judged as more natural when used with small set sizes than when used with large ones as determined in a model predicting mean naturalness rating from a continuous set size predictor (b = 0.05, SE = 0.01, p < .001). The exception is ten, which is judged to be slightly more natural than the surrounding terms. 3.3. Discussion The results of the two rating studies suggest that a listener’s perception of an expression’s naturalness is directly affected by the availability of lexical alternatives. With the


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exception of 6 and 7 (around half of the original set size), numbers are always judged to be more natural than some/summa when they are intermixed. This difference, however, is largest for the smallest set sizes. As predicted, the reduced naturalness of some/summa used with small sets, established in Experiment 1, decreased further when number terms were added to the stimuli. Therefore, at least in off-line judgments, listeners take into account what the speaker could have said, but did not. The results of Experiments 1 and 2 establish that the naturalness of descriptions with some varies with set size and for small set sizes is affected by the inclusion of number. Note, as mentioned in the Introduction, that these patterns are not predicted by the Literal-First and Default models. However, because these models focus on differences in the time-course of processing, these results cannot be taken as evidence against these models. Crucially, we use the obtained naturalness ratings to test the Default, Literal-First, and Constraint-Based model in Experiment 3, which evaluated competing predictions about time course using response times as the primary dependent measure.

4. Experiment 3 Experiment 3 was designed to test whether the effect of available natural alternatives is reflected in response times. Using the same paradigm and stimuli, we recorded participants’ judgments and response times to press one of two buttons (YES or NO) depending on whether they agreed or disagreed with the description. Based on the naturalness results from Experiments 1 and 2, the Constraint-Based account predicts that participants’ YES

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responses should be slower for more unnatural statements. Specifically, for some and summa response times are predicted to be slower compared to their more natural alternatives when used with (a) the unpartitioned set, where all is a more natural alternative and (b) in the small set range, where number terms are more natural and more rapidly available. Based on the naturalness data, the largest effect is expected for a set size of one, a somewhat smaller effect for two, and a still smaller effect for three. Response times for some/summa with these set sizes should be slower than when some and summa are used in the preferred range (four to seven gumballs). In Experiments 1 and 2, we also observed a difference in naturalness of simple versus partitive some for the unpartitioned set. This difference should be reflected both in the number of YES responses (more YES responses to some than to summa) and in response times (faster YES responses for some than for summa). Note that neither the Default nor the Literal-First model predicts response time differences based on naturalness of alternatives—regardless of set size, processing of a statement with some should take the same amount of time, except for the unpartitioned set, where the Default model predicts longer response times for semantic YES responses and the Literal-First model predicts longer response times for pragmatic NO responses. In addition, neither of these models predicts when a statement with some should result in a pragmatic NO judgment despite being semantically true. In contrast, the Constraint-Based account predicts the proportion of NO judgments to be proportional to the naturalness of some used with that set size. The conditions in which some/summa are used with the unpartitioned set are of additional interest because they can be linked to the literature using sentence verification tasks. In these conditions, enriching the statement to You got some but not all (of the) gumballs via scalar implicature makes it false. However, if no such pragmatic enrichment takes place, it is true. That is, YES responses reflect the semantic, at least, interpretation of the quantifier, whereas NO responses reflect the pragmatic, but not all, interpretation. Noveck and Posada (2003) and Bott and Noveck (2004) called the former logical responses and the latter pragmatic responses; we will make the same distinction but use the terms semantic and pragmatic and in addition characterize responders by how consistent they were in their responses. Analysis of semantic and pragmatic response times will allow us to test the predictions of the Constraint-Based, Literal-First, and Default account. In Bott and Noveck’s sentence verification paradigm, participants were asked to perform a two-alternative, forced-choice task. Participants were asked to respond TRUE or FALSE to clearly true, clearly false, and underinformative items (e.g., Some elephants have trunks). Bott and Noveck found that (a) pragmatic responses reflecting the implicature were slower than semantic responses; and (b) pragmatic responses were slower than TRUE responses to all for the unpartitioned set. If processing of the scalar item some proceeds similarly in our paradigm, we would expect to replicate Bott and Noveck’s pattern of results for the unpartitioned set with YES responses to some being faster than NO responses, and NO responses to some being slower than YES responses to all.


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4.1. Methods 4.1.1. Participants Forty-seven undergraduate students from the University of Rochester were paid $7.50 to participate. 4.1.2. Procedure and materials The procedure was the same as in Experiments 1 and 2, except that (a) participants heard a “ka-ching” sound before the gumballs moved from the upper to the lower chamber and (b) participants responded by pressing one of two buttons to indicate that YES, they agreed with, or NO, they disagreed with, the spoken description. Participants were asked to respond as quickly as possible. If they did not respond within 4 seconds of stimulus onset, the trial timed out and the next trial began. Participants’ judgments and response times were recorded. Participants were presented with the same types of stimuli as in Experiments 1 and 2. Because this experiment was conducted in a controlled laboratory setting rather than over the Web, we were able to gather more data from each participant. However, even in the laboratory setting we could not collect judgments from each participant for every quantifier/set size combination; that would have required 224 trials to collect a single data point for each quantifier/set size combination (14 set sizes and the 16 quantifiers from Experiment 2). Instead, we sampled a subset of the space of quantifier/set size combinations with each participant. Each participant received 136 trials. Of those, 80 were the same across participants and represented the quantifier/set size combinations that were of most interest (see Table 2). The remaining 56 trials were pseudo-randomly sampled combinations of quantifier and set size, with only one trial per sampled combination. Trials were sampled as follows. For some and summa, four set sizes were randomly sampled from the mid range (5–8) gumballs. For all, four set sizes were randomly sampled from 0 to 10 gumballs. For none, four set sizes were randomly sampled from 3 to 13 gumballs. For both one and two, four additional incorrect set sizes were sampled, one each from the small set range (1–4 gumballs, excluding the correct set size), the mid range (5–8 gumballs), the high range (9–12 Table 2 Distribution of the 80 trials each participant saw, over quantifiers, and set sizes Set Size Quantifier

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gumballs), and one of 0 or 13 gumballs. Finally, four additional number terms were sampled (one each) from the set of three or four, five to seven, eight or nine, and ten to twelve. This ensured that number terms were not all clustered at one end of the full range. Each number term occurred four times with its correct set size and four times with an incorrect number, one each sampled from the small set range, the mid range, the high range, and one of 0 or 13 gumballs, excluding the correct set size. For example, three could occur 4 times with 3 gumballs, once with 4, once with 7, once with 11, and once with 13 gumballs. The reason we included so many false number trials was to provide an approximate balance of YES and NO responses to avoid inducing an overall YES bias that might influence participants’ response times. To summarize, there were 28 some and summa trials each, 16 all trials, 16 none trials, 8 one trials, 8 two trials, and 8 trials each for four additional number terms. Of these, 64 were YES trials, 60 were NO trials, and 12 were critical trials—cases of some/summa used with the unpartitioned set, where they were underinformative, and some/summa used with one gumball, where a NO response is expected if the statement triggers a plural implicature (at least two gumballs, Zweig, 2009) and a YES response if it does not. Finally, there were three different versions of each image, with slightly different arrangements of gumballs to discourage participants from forming quantifier— image associations. 4.2. Results A total of 6,392 responses were recorded. Of those, 26 trials were excluded because participants did not respond within the 4 seconds provided before the trial timed out. These were mostly cases of high number terms occurring with big set sizes that required counting (e.g., 11 eleven trials, 8 twelve trials, 5 ten trials). Further, 33 cases with response times above or below 3 SDs from the grand mean of response times were also excluded. Finally, 254 cases of incorrect responses were excluded from the analysis. These were mostly cases of quantifier and set size combinations where counting a large set was necessary and the set size differed only slightly from the correct set size (e.g., ten used with a set size of 9). In total, 4.9% of the initial data set was excluded from the analysis. We organize the results as follows. We first report the proportion of YES and NO responses, focusing on the relationship between response choice and the naturalness ratings. These judgments will not allow us to tease apart the predictions of the Default and Literal-First model because these models do not make any predictions about response choices. However, they will allow us to test whether the naturalness differences between all, some, and summa are reflected in participants’ binary response choices. We then turn to the relationship between response times and naturalness ratings, testing the predictions we outlined earlier. Finally, we examine judgments and response times for pragmatic and semantic responses, relating our results to earlier work by Bott and Noveck (2004) and Noveck and Posada (2003). The results will be discussed with respect to the predictions made by the Default, Literal-First, and Constraint-Based models.


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4.2.1. Judgments All statistical analyses were obtained from mixed effects logistic regressions predicting the binary response outcome (YES or NO) from the predictors of interest. All models were fitted with the maximal random effects structure with by-participant random slopes for all within-participant factors of interest unless mentioned otherwise, following the guidelines in Barr, Levy, Scheepers, and Tily (2013). We first examine the unpartitioned some/summa conditions, which are functionally equivalent to the underinformative conditions from Noveck and Posada (2003) and Bott and Noveck (2004). Recall that under a semantic interpretation of some (You got some, and possibly all of the gumballs), participants should respond YES when they get all of the gumballs, while a pragmatic interpretation (You got some, but not all of the gumballs) yields a NO response. The judgment data qualitatively replicate the findings from the earlier studies: 100% of participants’ responses to all were YES, compared with 71% YES responses to partitive some. Judgments for simple some were intermediate between the two, with 82% YES responses. The difference between some and summa was significant in a mixed effects logistic regression predicting the binary response outcome (YES or NO) from quantifier (some or summa). The log odds of a YES response are lower for summa than some (b = 1.18, SE = 0.34, p < .001), reflecting the naturalness results obtained in Experiments 1 and 2, where summa was judged as less natural than some when used with the unpartitioned set. Thus, both the word some and its use in the partitive construction increase the probability and/or strength of generating an implicature, consistent with the naturalness-based predictions of the Constraint-Based account. 4.2.2. Response time analysis of naturalness effects on YES responses Response times ranged from 577 to 3,574 ms (mean: 1,420 ms, SD: 444 ms). Results of statistical analyses were obtained from mixed effects linear regression models predicting log-transformed response times from the predictors of interest. As with the judgment data, all models were fitted with the maximal random effects structure with by-participant random slopes for all within-participant factors of interest. Significance of predictors was confirmed by performing likelihood ratio tests, in which the deviance ( 2LL) of a model containing the fixed effect is compared to another model without that effect that is otherwise identical. This is one of the procedures recommended by Barr et al. (2013) for models containing random correlation parameters, as MCMC sampling (the approach recommended by Baayen et al., 2008) is not implemented in the R lme4 package. For YES responses at the unpartitioned set, quantifier was Helmert-coded. Two Helmert contrasts over the three levels of quantifier were included in the model, comparing each of the more natural levels against all less natural levels (all vs. some/summa, some vs. summa). YES responses to all were faster than some/summa (b = 0.26, SE = 0.02, t = 10.79) and YES responses to some were faster than to summa (b = 0.09, SE = 0.03, t = 3.24). YES responses to some and summa were slower for the unpartitioned set than in the most natural range determined in Experiments 1 and 2, four to seven gumballs (b = 0.09, SE = 0.03, t = 3.44).

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A similar pattern holds in the small set range for the comparison between some/summa and number terms: Responses to both some (b = 0.12, SE = 0.02, t = 5.7) and summa (b = 0.12, SE = 0.02, t = 5.8) were slower than number terms. Response times in the small set range did not differ for some and summa, as determined in model comparison between a model with and without a quantifier predictor (b = 0.02, SE = 0.02, t = 1.4), so we collapse them in further analysis. There was a main effect of set size in the small set range: Responses were faster as set size increased (b = 0.02, SE = 0.009, t = 2.54). The interaction between set size and quantifier (number term vs. some) was also significant (b = 0.08, SE = 0.02, t = 4.32), such that there was no difference in response times for number terms used with different set sizes in the subitizing range, but response times decreased for some/ summa with increasing set size. That is, the difference in response time between some/ summa and number terms is largest for one gumball, somewhat smaller for two gumballs, and smaller still for three gumballs. This mirrors the naturalness data obtained in Experiments 1 and 2 (see Fig. 6). Comparing response times for some/summa in the small set range to those in the preferred range (4–7), the results are similar: Responses in the small set range are slower than in the preferred range (b = 0.04, SE = 0.02, t = 2.3). Mean response times for YES responses are shown in Fig. 8 (response times for some and summa are collapsed as they did not differ). Analyzing the overall effect of naturalness on response times (not restricted to some/ summa) yields the following results. The Spearman r between log-transformed response times and mean naturalness for a given quantifier and set size combination was 0.1 for

2250 Mean response time (ms)

Quantifier exact

2000

some 1750

1500

1250

0

1

2

3

4

5

6 7 8 Set size

9

10 11 12 13

Fig. 8. Mean response times of YES responses to “some” (collapsed over simple and partitive use) and for exact quantifiers and number terms. For exact quantifiers, only the response time for their correct cardinality is plotted.


693

YES responses overall (collapsed over quantifier). This value increased to 0.3 upon exclusion of cases of number terms used outside the subitizing range, where counting is necessary to determine set size. This correlation was significant in a model predicting log-transformed response time from a centered naturalness predictor and a centered control predictor coding cases of number terms used outside the small set range as 1 and all other cases as 0. The main effect of naturalness was significant in the predicted direction, such that more natural combinations of quantifiers and set sizes were responded to more quickly than less natural ones (b = 0.04, SE = 0.01, t = 9.3). In addition, a main effect of the control predictor revealed that number terms used outside the subitizing range are responded to more slowly than cases that do not require counting (b = 0.44, SE = 0.02, t = 18.6). 4.2.3. Judgments and YES response times analyzed by pragmatic and semantic responders Table 3 shows the distribution of participants over number of semantic responses to some/summa at the unpartitioned set. Noveck and Posada (2003) and Bott and Noveck (2004) found that individual participants had a strong tendency to give mostly pragmatic or mostly semantic responses at the unpartitioned set.5 Therefore, they conducted subanalyses comparing pragmatic and semantic responders. Our participants were less consistent. Rather than two groups of responders clustered at either end of the full range, we observe a continuum in participants’ response behavior, with more participants clustered at the semantic end. Forty-two percent of the participants gave 100% (8) semantic responses. Dividing participants into two groups, semantic and pragmatic responders, where pragmatic responders are defined as those participants who gave pragmatic responses more than half of the time and semantic responders as those who responded semantically more than half of the time yields a large group of semantic responders (38 participants, 81%) and a smaller group of pragmatic responders (7 participants, 15%). Two participants (4%) gave an equal number of semantic and pragmatic responses. Given the nature of the distribution, rather than analyzing response times for semantic and pragmatic responders separately, we included a continuous predictor of responder type (degree of “semanticity” as determined by number of semantic responses) as a control variable in the analyses.6 That is, a participant with one semantic response was treated as a more pragmatic responder than a participant with five semantic responses. We analyzed the effect of (continuous) responder type on the response time effects reported above, specifically (a) the naturalness effect at the unpartitioned set and (b) the naturalness effect for small sets. First, we included centered continuous responder type as interaction terms with the Helmert contrasts for quantifier (all vs. some/summa, some vs. summa) for the unpartitioned set. In this model, both interactions were significant: the Table 3 Distribution of participants over number of semantic responses given Number of semantic responses Number of participants

0 2

1 2

2 2

3 1

4 2

5 5

6 6

7 6

8 21

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difference between some and summa was more pronounced for more pragmatic responders (b = 0.05, SE = 0.02, t = 2.5), while the difference between all and some/summa was significantly different for different responder types (b = 0.04, SE = 0.01, t = 3.97) but seems to be better accounted for by participants’ response consistency (see below). This suggests that more pragmatic responders are more sensitive to the relative naturalness of simple and partitive some used with an unpartitioned set. We return to this finding in the discussion. An analysis of responder type for the naturalness effects for small set sizes yielded no significant results. That is, responder type did not interact with the quantifier by set size interaction reported above. 4.2.4. Response time analysis for semantic versus pragmatic responses to some/summa Recall that the Default model predicts pragmatic NO responses to some/summa with the unpartitioned set to be faster than semantic YES responses, while the reverse is the case for the Literal-First hypothesis. The latter has found support in a similar sentence verification task as the one reported here (Bott & Noveck, 2004). We thus attempted to replicate Bott and Noveck’s finding that pragmatic NO responses are slower than semantic YES responses. To this end we conducted four different analyses. All were linear regression models predicting log-transformed response time from response and quantifier predictors and their interaction (the interaction terms were included to test for whether NO and YES responses were arrived at with different speed for some and summa): In model 1, we compared all YES responses to all NO responses. Then we performed the Bott and Noveck between-participants analysis comparing only responses from participants who responded entirely consistently to some/summa (i.e., either 8 or 0 semantic responses in total) in model 2. In a very similar between-participants analysis, we compared response times from participants who responded entirely consistently to either some or summa (i.e., either 4 or 0 semantic responses to either quantifier) in model 3. Finally, again following Bott and Noveck, we compared response times to YES and NO responses within participants, excluding the consistent responders that entered model 2 from model 4. Results are summarized in Table 4. The interaction between quantifier and response was not significant in any of the models, suggesting there was no difference between some and summa in the speed with which participants responded YES or NO. The main effect of quantifier reached significance in both model 1 (all responses to some/summa at unpartitioned set) and model 4 (including only inconsistent responders), such that responses to summa were generally slower than those to some (see Table 4 for coefficients). Finally, the main effect of response was marginally significant in models 2 and 3 (including only consistent responders, either overall, or within quantifier condition), such that YES responses were marginally faster than NO responses. 4.2.5. Response times as a function of response inconsistency We conducted a final response time analysis that was motivated by the overall inconsistency in participants’ response behavior at the unpartitioned set. Rather than analyzing


695

Table 4 Model coefficients, standard error, t-value, and p-value for the three predictors (quantifier, response, and their interaction) in each of four different models Quantifier (some, summa) Model

Obs.

b

SE

t

Quantifier : Response Interaction

Response (no, yes) p

b

SE

t

p

b

SE

t

p

1. Overall 375 0.06 0.02 2.69

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